where \(\Delta \Omega \)  is the space angle obtained from the angular separation \(\Delta \Psi \) , and \(N_{\rm T}\)  is the total number of GRBs that satisfy the time criterion Eq. (REF ) of neutrino events. Here we count all GRBs collected in the GRB catalog [1]}.
The localization information of the GRBs that associated with 9 shower events with the energy from 60 to 500 TeV can be found in
refs. [2]}, [3]}. The directional information of 4 events with the
energy above PeV can be found in ref. [4]}. Actually the detail
information of all these GRBs can be found in the GRB catalogs
that are collected on the website [1]} http://www.mpe.mpg.de/ jcg/grblink.html, so as the GRBs associated with track events which are proposed in this work.
The background estimations are also listed in Table REF . We find that most of GRB candidates stand out from the background, and only 2 candidates have background errors larger than 50%. The sum of \(N_{\rm B}\)  is 3.76, which is less than 30% of the total number. Furthermore, if regarding the 2 large background candidates as the statistical errors and excluding them, we can get the total background GRB number \(\Sigma ^{\prime }N_{\rm B}=1.084\)  for the rest 10 candidates. Therefore, these 10 candidates are approximately 10 times higher than the background value.
